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Theorem isfin2 8663
Description: Definition of a II-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
Assertion
Ref Expression
isfin2  |-  ( A  e.  V  ->  ( A  e. FinII 
<-> 
A. y  e.  ~P  ~P A ( ( y  =/=  (/)  /\ [ C.]  Or  y
)  ->  U. y  e.  y ) ) )
Distinct variable group:    y, A
Allowed substitution hint:    V( y)

Proof of Theorem isfin2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 pweq 4006 . . . 4  |-  ( x  =  A  ->  ~P x  =  ~P A
)
21pweqd 4008 . . 3  |-  ( x  =  A  ->  ~P ~P x  =  ~P ~P A )
32raleqdv 3057 . 2  |-  ( x  =  A  ->  ( A. y  e.  ~P  ~P x ( ( y  =/=  (/)  /\ [ C.]  Or  y
)  ->  U. y  e.  y )  <->  A. y  e.  ~P  ~P A ( ( y  =/=  (/)  /\ [ C.]  Or  y )  ->  U. y  e.  y ) ) )
4 df-fin2 8655 . 2  |- FinII  =  {
x  |  A. y  e.  ~P  ~P x ( ( y  =/=  (/)  /\ [ C.]  Or  y )  ->  U. y  e.  y ) }
53, 4elab2g 3245 1  |-  ( A  e.  V  ->  ( A  e. FinII 
<-> 
A. y  e.  ~P  ~P A ( ( y  =/=  (/)  /\ [ C.]  Or  y
)  ->  U. y  e.  y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2655   A.wral 2807   (/)c0 3778   ~Pcpw 4003   U.cuni 4238    Or wor 4792   [ C.] crpss 6554  FinIIcfin2 8648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ral 2812  df-v 3108  df-in 3476  df-ss 3483  df-pw 4005  df-fin2 8655
This theorem is referenced by:  fin2i  8664  isfin2-2  8688  ssfin2  8689  enfin2i  8690  fin12  8782  fin1a2s  8783
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